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A152668
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Number of runs of even entries in all permutations of {1,2,...,n} (the permutation 274831659 has 3 runs of even entries: 2, 48 and 6).
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1
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2, 6, 36, 192, 1440, 10800, 100800, 967680, 10886400, 127008000, 1676505600, 22992076800, 348713164800, 5492232345600, 94152554496000, 1673823191040000, 32011868528640000, 633834996867072000, 13380961044971520000
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OFFSET
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2,1
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COMMENTS
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a(n) = Sum(k*A152667(n,k), k=1..floor(n/2)).
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LINKS
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FORMULA
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a(2n) = (n+1)(2n)!/2;
a(2n+1) = n(n+2)(2n)!.
D-finite with recurrence a(n) -2*a(n-1) -n*(n-1)*a(n-2) +2*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 24 2022
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EXAMPLE
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a(3) = 6 because each of the permutations 123, 132, 213, 231, 312, 321 has exactly 1 run of even entries.
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MAPLE
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ae := proc (n) options operator, arrow: (1/2)*factorial(2*n)*(n+1) end proc: ao := proc (n) options operator, arrow: n*(n+2)*factorial(2*n) end proc: a := proc (n) if `mod`(n, 2) = 0 then ae((1/2)*n) else ao((1/2)*n-1/2) end if end proc; seq(a(n), n = 2 .. 20);
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MATHEMATICA
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a[n_] := If[EvenQ[n], (n/2+1)n!/2, ((n-1)/2)((n-1)/2+2)(n-1)!];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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